On Inefficiency of Markowitz-Style Investment Strategies When Drawdown is Important

The focal point of this paper is the issue of "drawdown" which arises in recursive betting scenarios and related applications in the stock market. Roughly speaking, drawdown is understood to mean drops in wealth over time from peaks to subsequent lows. Motivated by the fact that this issue is of paramount concern to conservative investors, we dispense with the classical variance as the risk metric and work with drawdown and mean return as the risk-reward pair. In this setting, the main results in this paper address the so-called "efficiency" of linear time-invariant (LTI) investment feedback strategies which correspond to Markowitz-style schemes in the finance literature. Our analysis begins with the following principle which is widely used in finance: Given two investment opportunities, if one of them has higher risk and lower return, it will be deemed to be inefficient or strictly dominated and generally rejected in the marketplace. In this framework, with risk-reward pair as described above, our main result is that classical Markowitz-style strategies are inefficient. To establish this, we use a new investment strategy which involves a time-varying linear feedback block K(k), called the drawdown modulator. Using this instead of the original LTI feedback block K in the Markowitz scheme, the desired domination is obtained. As a bonus, it is also seen that the modulator assures a worst-case level of drawdown protection with probability one.Comment: This paper has been published in Proceedings of 56th IEEE Conference on Decision and Control (CDC) 201

Ekspektasi Maksimum Percentage Drawdown pada data Saham PT. Mayora Tbk dengan simulasi Monte Carlo

A drawdown is a tool for defining trading strategies for commodities, stocks, and investments. This analysis is one way of monitoring the decline in asset value over a certain period of time. This journal will discuss PT.MayoraTbk stock trading strategy. By analyzing the observed drawdown in the specified time period. The drawdown analysis here uses the feedback control on PT.MayoraTbk stock trading is assumed to follow the geometric Brownian motion. The data obtained is tested whether the data meets Brown's motion assumptions. Then the maximum drawdown expectation is determined at the selected time interval. An estimate is carried out for the maximum expected drawdown percentage of the share value. To test the validity of the estimation results, a Monte Carlo simulation is carried out. Monte Carlo simulation with the term Sampling Simulation or Monte Carlo Sampling Technique. This simulation sampling illustrates the possible use of sample data using the Monte Carlo method and also the distribution can be known or estimated. This simulation uses existing data (historical data) that is actually used in a simulation that includes inventory or sampling with a known and determined probability distribution, so this Monte Carlo simulation can be used. The basic idea of this Monte Carlo simulation is to generate or generate a value to form a model of the variables and study it

On Some Stochastic Optimal Control Problems in Actuarial Mathematics

The event of ruin (bankruptcy) has long been a core concept of risk management interest in the literature of actuarial science. There are two major research lines. The first one focuses on distributional studies of some crucial ruin-related variables such as the deficit at ruin or the time to ruin. The second one focuses on dynamically controlling the probability that ruin occurs by imposing controls such as investment, reinsurance, or dividend payouts. The content of the thesis will be in line with the second research direction, but under a relaxed definition of ruin, for the reason that ruin is often too harsh a criteria to be implemented in practice. Relaxation of the concept of ruin through the consideration of "exotic ruin" features, including for instance, ruin under discrete observations, Parisian ruin setup, two-sided exit framework, and drawdown setup, received considerable attention in recent years. While there has been a rich literature on the distributional studies of those new features in insurance surplus processes, comparably less contributions have been made to dynamically controlling the corresponding risk. The thesis proposes to analytically study stochastic control problems related to some "exotic ruin" features in the broad area of insurance and finance. In particular, in Chapter 3, we study an optimal investment problem by minimizing the probability that a significant drawdown occurs. In Chapter 4, we take this analysis one step further by proposing a general drawdown-based penalty structure, which include for example, the probability of drawdown considered in Chapter 3 as a special case. Subsequently, we apply it in an optimal investment problem of maximizing a fund manager's expected cumulative income. Moreover, in Chapter 5 we study an optimal investment-reinsurance problem in a two-sided exit framework. All problems mentioned above are considered in a random time horizon. Although the random time horizon is mainly determined by the nature of the problem, we point out that under suitable assumptions, a random time horizon is analytically more tractable in comparison to its finite deterministic counterpart. For each problem considered in Chapters 3--5, we will adopt the dynamic programming principle (DPP) to derive a partial differential equation (PDE), commonly referred to as a Hamilton-Jacobi-Bellman (HJB) equation in the literature, and subsequently show that the value function of each problem is equivalent to a strong solution to the associated HJB equation via a verification argument. The remaining problem is then to solve the HJB equations explicitly. We will develop a new decomposition method in Chapter 3, which decomposes a nonlinear second-order ordinary differential equation (ODE) into two solvable nonlinear first-order ODEs. In Chapters 4 and 5, we use the Legendre transform to build respectively one-to-one correspondence between the original problem and its dual problem, with the latter being a linear free boundary problem that can be solved in explicit forms. It is worth mentioning that additional difficulties arise in the drawdown related problems of Chapters 3 and 4 for the reason that the underlying problems involve the maximum process as an additional dimension. We overcome this difficulty by utilizing a dimension reduction technique. Chapter 6 will be devoted to the study of an optimal investment-reinsurance problem of maximizing the expected mean-variance utility function, which is a typical time-inconsistent problem in the sense that DPP fails. The problem is then formulated as a non-cooperative game, and a subgame perfect Nash equilibrium is subsequently solved. The thesis is finally ended with some concluding remarks and some future research directions in Chapter 7

Critical Market Crashes

This review is a partial synthesis of the book ``Why stock market crash'' (Princeton University Press, January 2003), which presents a general theory of financial crashes and of stock market instabilities that his co-workers and the author have developed over the past seven years. The study of the frequency distribution of drawdowns, or runs of successive losses shows that large financial crashes are ``outliers'': they form a class of their own as can be seen from their statistical signatures. If large financial crashes are ``outliers'', they are special and thus require a special explanation, a specific model, a theory of their own. In addition, their special properties may perhaps be used for their prediction. The main mechanisms leading to positive feedbacks, i.e., self-reinforcement, such as imitative behavior and herding between investors are reviewed with many references provided to the relevant literature outside the confine of Physics. Positive feedbacks provide the fuel for the development of speculative bubbles, preparing the instability for a major crash. We demonstrate several detailed mathematical models of speculative bubbles and crashes. The most important message is the discovery of robust and universal signatures of the approach to crashes. These precursory patterns have been documented for essentially all crashes on developed as well as emergent stock markets, on currency markets, on company stocks, and so on. The concept of an ``anti-bubble'' is also summarized, with two forward predictions on the Japanese stock market starting in 1999 and on the USA stock market still running. We conclude by presenting our view of the organization of financial markets.Comment: Latex 89 pages and 38 figures, in press in Physics Report

Generating drawdown-realistic financial price paths using path signatures

A novel generative machine learning approach for the simulation of sequences of financial price data with drawdowns quantifiably close to empirical data is introduced. Applications such as pricing drawdown insurance options or developing portfolio drawdown control strategies call for a host of drawdown-realistic paths. Historical scenarios may be insufficient to effectively train and backtest the strategy, while standard parametric Monte Carlo does not adequately preserve drawdowns. We advocate a non-parametric Monte Carlo approach combining a variational autoencoder generative model with a drawdown reconstruction loss function. To overcome issues of numerical complexity and non-differentiability, we approximate drawdown as a linear function of the moments of the path, known in the literature as path signatures. We prove the required regularity of drawdown function and consistency of the approximation. Furthermore, we obtain close numerical approximations using linear regression for fractional Brownian and empirical data. We argue that linear combinations of the moments of a path yield a mathematically non-trivial smoothing of the drawdown function, which gives one leeway to simulate drawdown-realistic price paths by including drawdown evaluation metrics in the learning objective. We conclude with numerical experiments on mixed equity, bond, real estate and commodity portfolios and obtain a host of drawdown-realistic paths